Text of Relativistic Quantum Chemistry Applied to Actinides
by
A Thesis submitted to the Faculty of Graduate Studies of
The University of Manitoba
in partial fulfillment of the requirements of the degree of
DOCTOR OF PHILOSOPHY
Department of Chemistry
University of Manitoba
i
1.2. Chemical Properties of Actinides
……………...………………..…………………………..4
1.3. Theoretical Studies of Actinide Complexes
….…………………..…………………….…...9
1.3.1. The Schrödinger Equation
….…………………..………………..…………….………….9
1.3.2. The Variational Principle and Electronic Basis
Sets….………….……………………….10
1.3.3. The Hartree-Fock Method and Post Hartree-Fock Approaches
…..…………...…………13
1.3.4. Density Functional Theory
….…………………………………….………….…………..17
1.3.5. Relativistic Effects ….……………………………….……………….…………………..21
1.3.6. Solvation Effects ….………………………………….…………….…………………….25
1.5. References
....…….…………………………...…….……………………....……………....29
Chapter 2- Performance of Relativistic Effective Core Potentials
for DFT Calculations on
Actinide Compounds ….……………………………………….……………….………………36
Chapter 3- Theoretical Study of the Structural Properties of
Plutonium IV and VI Complexes
….……………………………………….……………………………………….……………...64
Chapter 4- Theoretical Study of the Structural and Electronic
Properties of Plutonyl Hydroxides
….……………………………………….………………………………………………………98
Chapter 5- DFT Study of Uranyl Peroxo Complexes with H2O, F - ,
OH
- , CO3
- ….151
Chapter 6- Theoretical Study of the Reduction of Uranium(VI) Aquo
Complexes on Titania
Particles and by Alcohols …………………….……….………..……………………………...195
Chapter 7- QM and QM/MM Study of Uranyl Fluorides in the Gas Phase,
Aqueous Phase and in
the Hydrophobic Cavities of Tetrabrachion
…………………………………………………...231
Chapter 8- Novel and Stable U(V)/U(V) Binuclear Complexes Formed by
Oxo-functionalization
of Axial Oxo Atoms …………………………………………………………………………..269
Chapter 9- Theoretical Study of a Gas-Phase Binuclear Uranyl
Hydroxo Complex,
[(UO2)2(OH)5] - …………………………….………………………………………………….295
Chapter 10- Summary and Future Studies of Actinide Complexes
…………………………..325
10.1. Summary ………………………………………………………………………………...325
10.2. Future Studies of Actinide Complexes Specific to this Thesis
…………..……………...328
10.3. General Directions for Computational Studies of Actinide
Complexes ………………...330
References ……………….……………….……………….……………….…………………..331
List of Figures
Figure 1.1: The actinides in the periodic table of elements
……………………………..……….2
Figure 1.2: The aquo complexes of U(VI) and U(IV)
…………………………………..………8
Figure 1.3: Schematic description of the chapters in this thesis
………………………….…….28
Figure 2.1: Absolute Deviations of the Vibrational Wavenumbers, cm
-1
for NUN, NUO + and
CUO computed using Stuttgart RECPs with the aug-cc-pVTZ ligand
basis set and the AE four-
component method with the L2 basis set
………………………………………………….…...44
Figure 2.2: Distorted Planar T-Shaped Structure of UO3(g)
……………………………….…..46
Figure 2.3: Absolute Deviations of the Vibrational Wavenumbers, cm
-1
, for UO3 and UF6
computed using Stuttgart RECPs with the aug-cc-pVTZ ligand basis
set and the AE four-
component method with the L2 basis set
………………………………………………….…..50
Figure 2.4: Absolute Deviations of the Enthalpy Changes, kJ/mol for
Reactions 1-5 computed
using Stuttgart RECPs with the aug-cc-pVTZ ligand basis set and the
AE four-component
method …………………………………………………………………………………….…...55
Figure 3.1: (Left) The An=Oyl bond lengths (Å) in the bare actinyl
moieties, AnO2 2+
as well as
.
) in the
and the pentaaquo complexes, [AnO2(H2O)5] 2+
. ………….…...74
Figure 3.2: (Left) The An=Oyl bond lengths (Å) in the bare actinyl
moieties, AnO2 1+
as well as
the An=Oyl bond lengths and average An-OH2 bond lengths (Å) in the
pentaaquo complexes,
v
frequencies (cm -1
and the pentaaquo complexes,
[AnO2(H2O)5] 1+
. These values were obtained with the obtained using the PBE
functional and
PCM solvation model ……………………………………………………………………….….75
Figure 3.3: A. The cis (top) and trans (bottom) isomers of the
PuO2Cl2(H2O)2 complex. B. The
cis (top) and trans (bottom) isomers of the PuO2Cl2(H2O)3 complex
…………………………80
Figure 3.4: A) The decrease in the An=Oyl and An-Cl bond lengths
(Å) on addition of 2Cs + to the
[AnO2Cl4] 2-
)
2- complexes …………………………………………….81
using the B3LYP functional ……………………………………………………………………85
Figure 3.6: The optimized structures of several plutonium (IV)
complexes: A) Pu(H2O)8 4+
, B)
calculated with the B3LYP functional in the
aqueous phase …………………………………………………………………………………..89
complexes
obtained at the BP86/B2 level with the COSMO solvation model
……………..……….……105
Figure 4.2: Optimized structures of the aquo-hydroxo
[PuO2(H2O)4-n(OH)n] 2-n
complexes
obtained at the BP86/B2 level with the COSMO solvation model
…………………………...106
Figure 4.3: Optimized structures of the bis-plutonyl aquo
tetrahydroxo and dihydroxo complexes
obtained at the B3LYP/B3 level in the gas phase
…………...………………………………..114
vi
Figure 4.4: Structures of [(PuO2)3(H2O)6(O)(OH)3] + obtained at the
BP86/B2/COSMO level. The
C3V structure is shown on the left while the structure with a
bridging aquo group is shown on the
right ……………………………………………………………………………………………115
Figure 4.5: Scheme depicting the formation and decomposition of the
μ3-oxo hexagonal
trimetallic core of the trimer aquo-hydroxo complexes
………….…………………………....127
Figure 4.6: The molecular orbitals of [PuO2] 2+
...…………….……..……………...………....129
optimized at the BP86/B2 level with the COSMO
model ………………………………………………………………………………………….130
Figure 4.8: Selected occupied alpha spin MOs of [PuO2(OH)4]
2-
……………………..……..132
optimized at the BP86/B2 level with the
COSMO model ……………………………………………….……………………………….134
Figure 4.10: The 39th to 62nd MOs of [(PuO2)2(OH)2] 2+
of alpha spin ……………..……….136
Figure 4.11: Selected alpha spin MOs of the trimer complex,
[(PuO2)3(O)(OH)3] + …..……...139
Figure 4.12: Abbreviated energy level diagram of the
[(AnO2)3(O)(OH)3] +
complexes ……...142
and its peroxo derivatives optimized at the B3LYP/B1 level
in aqueous solution ……………………………………….………………...…………………156
Figure 5.2: The molecular orbitals of UO2(O2). The geometry of this
complex was optimized
with the PCM approach and the B3LYP functional
………………………………………….159
vii
and its peroxo derivatives obtained at the at the
B3LYP/B1 level in aqueous solution ………………………………………………………….160
Figure 5.4: MO-26 and MO-27 of the C2v structure of UO2(O2)2
2-
………………………….163
and its peroxo derivatives optimized at the B3LYP/B1
level in aqueous solution ………………………………………………………………………166
Figure 5.6: The structures of UO2F4 2-
and its peroxo derivatives optimized at the B3LYP/B1
level in aqueous solution ………………………………………………………………………170
Figure 5.7: The structures of UO2(OH)4 2-
and its peroxo derivatives optimized at the B3LYP/B1
level in aqueous solution ………….……………………………………...……………………173
Figure 5.8: MO-30 of cis and trans- UO2(O2)2(OH)2 4-
………….………..…………………..176
and its peroxo derivatives optimized at the B3LYP/B1
level in aqueous solution ……………………………………………………………………....178
Figure 6.1: Possibilities for charge transfer to U(VI) complexes
adsorbed on TiO2 crystals or
nanoparticles …………………………………………………………………………………..199
Figure 6.2: Structure of UO2 2+
adsorbed at the rutile (110) surface at the bridging oxygen
atoms
…………………………...............…………………………………………………………….204
Figure 6.3: Total and partial electronic density of states (DOS)
obtained for a clean rutile (110)
slab while using with the PBE functional (left) and the PBE+U
approach (right, U=4.2 eV)
…………………………………………………………………………………………………206
viii
Figure 6.4: Total and partial electronic density of states (DOS)
obtained for hydroxylated rutile
(110) slabs with the PBE+U approach (U=4.2 eV for Ti 3d and U=4.0
eV for U 5f)
……………………………………….………………………………………………………...207
Figure 6.5: Total and partial electronic density of states (DOS)
obtained for rutile (110) slabs
with an adsorbed UO2 2+
group obtained with the PBE+U approach (U=4.2 eV for Ti-3d
and
U=4.0 eV for U-5f)….………………………………………………………………………...210
Figure 6.6: Total and partial electronic density of states (DOS)
obtained for stoichiometric (left)
and surface hydroxyl defected (right) rutile (110) slabs with
adsorbed [UO2(H2O)3] 2+
obtained
with the PBE+U approach (U=4.2 eV for Ti-3d and U=4.0 eV for U-5f)
………………….211
Figure 6.7: Electronic energy levels and frontier molecular
orbitals obtained at the BP86/TZP
level for the a Ti38076-[UO2(H2O)3] 2+
surface-adsorbate complex …………………………...212
Figure 6.8: Calculated spin distributions of a rutile (110) slab
with a surface-adsorbed
[UO2(H2O)3] 2+
moiety and a surface hydroxyl defect. These calculations were
carried out with
the PBE+U approach …………………………………………………………………………215
Figure 6.9: Electronic energy levels and MOs of a
non-stoichiometric cluster (with a surface
hydroxyl)-adsorbate complex …….......………………………………………………………216
Figure 6.10: The quenching of the lowest triplet excited state of
[UO2(H2O)5] 2+
by water and
organic alcohols ………………………….…………………………………………………...221
Figure 7.1: Top: the right handed coiled coil protein of
Tetrabrachion. Bottom: two monomer
chains of the RHCC tetramer ………………………………………………………………...233
ix
complexes optimized at the
BP86/TZP/ZORA/COSMO level ……………………………………………………………..240
Figure 7.3: Left: Variation of the Mulliken charges on the uranium
atom (black) and uranyl
moiety (red) and the fluoro-2p contribution to the HOMO-1 orbital
(blue) with increasing
number of fluoride ligands. Right: Variation of the U=O bond
lengths (black) and bond orders
(red) with increasing number of fluoride ligands
……………………………………………..246
Figure 7.4: Selected frontier molecular orbitals of [UO2Fn(H2O)5-n]
2-n
complexes. The member
structures were optimized at the ADF/ZORA/BP86/TZP/COSMO
……………….………....248
Figure 7.5: The atoms of the isoleucine residues (represented as
green crosses) at the n-terminal
end of cavity two ....…………………………………………………………………………..255
Figure 8.1: Two common motifs for cation-cation interactions
between uranyl groups ..........272
Figure 8.2: The recently synthesized U2O4 Pacman complex, 1a
…………............................273
Figure 8.3: Optimized structure of the yttrium dimer complex
obtained at the PBE/L1 level. The
O, U and Y atoms are in red, dark green and light green colours
respectively ………………274
Figure 8.4: The lithium-chloride monomer salt of the yttrium dimer
complex, 2. The dioxo-
uranium core with Li and Y oxo-functionalization are shown on the
right ….………………275
Figure 8.5: Molecular orbitals of primary σ- and π- character in
the unrestricted singlet state of
1a ….………………………………………………………………………………………….280
Figure 8.6: The (a) HOMO-172 (bonding with respect to the two U
atoms) with an energy of -
1.094 a.u. and (b) its antibonding counterpart, both containing
contributions from U-5f and O-2s
orbitals ………………………………………………………………………………………..281
x
Figure 9.1: The three low energy structures of (UO2)2(OH)5 -
obtained at the B3LYP/TZVP level
………………………………………………………………………………………………...301
Figure 9.2: Calculated IR spectra for the μ2-dihydroxo (black),
μ-hydroxo-CCI (blue) and μ-
hydroxo-di-CCI (red) structures of (UO2)2(OH)5 - obtained using the
B3LYP functional
………………………………………………………………………………………………...308
Figure 9.3: Other possible structure of (UO2)2(OH)5 - . Their
energies at the B3LYP (and MP2
//B3LYP) levels are given relative to that of the μ-hydroxo-di-CCI
structure
………………………………………………………………………………………………...313
Figure 9.4: Low energy structures of the bis-uranyl hydroxo
complexes, (UO2)2(OH)n 4-n
………………………………………………………………………………………………...316
xi
Table 1.1: Electronic ground state configurations of the actinide
elements ……………………..5
Table 1.2: The various oxidation states of the actinide elements
………………………………..6
Table 2.1: Vibrational Frequencies, cm -1
, of NUN computed using Stuttgart RECPs with the aug-
cc-pVTZ ligand basis set and the AE four-component method with the
L2 basis set …………42
Table 2.2: Vibrational Frequencies, cm -1
, of NUO + computed using Stuttgart RECPs with the
aug-cc-pVTZ ligand basis set and the AE four-component method with
the L2 basis set….….43
Table 2.3: Vibrational Frequencies, cm -1
, of CUO computed using Stuttgart RECPs with the aug-
cc-pVTZ ligand basis set and the AE four-component method with the
L2 basis set ………...45
Table 2.4: Calculated Bond Lengths (in ) and Bond Angles (in
degrees) of UO3 (g) computed
using Stuttgart RECPs with the aug-cc-pVTZ ligand basis set and the
AE four-component
method with the L2 basis set …………………………………………………………………...48
Table 2.5: Calculated Bond Lengths (in ) and Bond Angles (in
degrees) of UO3 (g) computed
using Stuttgart RECPs with the aug-cc-pVTZ ligand basis set and the
AE four-component
method with the L2 basis set ………………………………………………..…………….……49
Table 2.6: Vibrational Frequencies, cm -1
, of UF6 computed using Stuttgart RECPs with the aug-
cc-pVTZ ligand basis set and the AE four-component method with the
L2 basis set …………51
Table 2.7: Calculated enthalpy changes * , kJ/mol for Reactions 1-5
computed using Stuttgart
RECPs and the AE four-component method
…………………………………………………..53
xii
Table 3.1: The calculated bond lengths (Å) and plutonyl vibrational
frequencies (cm -1
) of the
systems obtained using DFT in the gaseous and
aqueous phases ………………………………………………………………………………….71
Table 3.2: The calculated structural properties (bond lengths in Å
and vibrational frequencies in
cm -1
) of the chloride complexes of the plutonyl (VI) cation obtained
using the B3LYP functional.
The values obtained with the PBE functional are given in
parenthesis ...………………………77
Table 3.3. The calculated structural properties (bond lengths in Å
and vibrational frequencies in
cm -1
) of the nitrate complexes of the plutonyl (VI) cation obtained
using the B3LYP functional
in the gaseous and aqueous phases. The values obtained with the PBE
functional are given in
parenthesis ………………………………..…………………………….……….………………84
Table 3.4: The calculated bond lengths (Å) of the AnO2(NO3)2(H2O)2,
AnO2(NO3)2(TBP)2 a ,
AnO2Cl2(H2O)2 and AnO2Cl2(TPPO)2 a complexes obtained using the
B3LYP functional with
RECPs in the gaseous phase ……………………………………………………………………86
Table 3.5: The calculated Pu-OH2 and Pu-Owater/nitrate bond lengths
of the aquo, nitrate and aquo-
nitrate complexes of the plutonium (IV) cation obtained using DFT
in the gaseous and aqueous
phases …………………………………………………………...………………………………88
and [PuO2(H2O)4(OH)] + . The bond
distances are given in Å while the vibrational frequencies
(asymmetric/symmetric plutonyl
stretching modes) are given in cm -1
…………...……………………………..………………..106
xiii
bond distances are given in Å while the vibrational frequencies
(asymmetric/symmetric plutonyl
stretching modes) are given in cm -1
…………………………………………………………...108
bond distances are given in Å while the vibrational frequencies
(asymmetric/symmetric plutonyl
stretching modes) are given in cm -1
…………………………………………………………...109
and [PuO2(OH)5] 3-
distances are given in Å while the vibrational frequencies
(asymmetric/symmetric plutonyl
stretching modes) are given in cm -1
…………………………………………………………...110
Table 4.5: Calculated Mayer bond orders of the plutonyl
aquo-hydroxo complexes obtained at
the PBE/B1 level in the gaseous phase
…..……….…………………………………………...111
Table 4.6: Calculated structural properties of the plutonyl dimer
complexes. The bond lengths
are in Å and the calculated IR intensities (km/mol) are given in
parenthesis …………………113
Table 4.7: Calculated structural properties of the μ3-oxo motifs of
the trimeric complex,
[(PuO2)3(H2O)6(O)(OH)3] + . The bond lengths are given in Å, bond
angles in degrees while
vibrational frequencies are presented in cm -1
…………………..……………………………..115
) of uranium aquo hydroxo
complexes obtained at the PBE/B1 level
……………………………………………………...118
Table 4.9: Calculated reaction energies (kcal/mol) for the
formation of the dimer and trimer
actinyl hydroxo complexes in the aqueous phase
……………………………………………..122
xiv
BP86/B2 while using the COSMO solvation model
………………………………………….124
Table 4.11: Energies, atomic contributions and descriptions of the
MOs of [(PuO2)2(OH)2] 2+
at
the B3LYP/B3 level. The orbital energies are scaled such that the
HOMO is at 0.00 eV …...137
Table 4.12: Energies (eV), individual atomic contributions and
descriptions of the MOs of
[(PuO2)3(O)(OH)3] + at the B3LYP/B3 level. The orbital energies are
scaled such that the HOMO
is at 0.00 eV …………………………………………………………………………………...141
Table 5.1: Energies and characters of the MOs of the dioxouranium
(VI) peroxides in aqueous
solution obtained at the B3LYP/B1 level. MO energies are given in
eV …………………….158
Table 5.2: Calculated structural properties and vibrational
frequencies of UO2 2+
and its peroxo
Table 5.3: Calculated structural properties and vibrational
frequencies (cm -1
) of UO2(H2O)5 2+
and
its peroxo derivatives obtained at the B3LYP/B1 level in the
gas-phase and in aqueous solution
…………………………………………………………………………………………………165
Table 5.4: Calculated structural properties and vibrational
frequencies of UO2F4 2-
and its peroxo
derivatives obtained at the B3LYP/B1 level in aqueous solution
…………………………….168
Table 5.5: Calculated structural properties and vibrational
frequencies of UO2(OH)4 2-
and its
peroxo derivatives obtained at the B3LYP/B1 level in aqueous
solution …………………….172
Table 5.6: Calculated structural properties and vibrational
frequencies (cm -1
) of UO2(CO3)3 4-
and
its peroxo derivatives obtained at the B3LYP/B1 level in the
gas-phase and in aqueous solution
…………………………………………………………………………………………………180
xv
Table 5.7: Calculated structural properties and vibrational
frequencies (cm -1
) of UO2(NO3)3 - and
its peroxo derivatives obtained at the B3LYP/B1 level in the
gas-phase and in aqueous solution
…………………………………………………………………………………………………181
Table 5.8: The calculated Mayer bond orders in various uranyl
complexes and their peroxo
derivatives obtained at the B3LYP/B2 level using structures
optimized at the B3LYP/B1 level.
……………………………………………………………………………………………..…..184
Table 6.1: Calculated structural features and properties of U(VI)
complexes obtained with the
PBE functional in the gas phase
……………………………………………………………....203
Table 6.2: Calculated reaction energies (kcal/mol) obtained for R6
in aqueous solution obtained
at the PBE/TZP and B3LYP/TZP levels in addition to the calculated
structural and electronic
properties of the triplet state uranyl-quencher complexes
……………………………………219
Table 7.1: Calculated and experimental structural parameters of the
[UO2Fn(H2O)5-n] 2-n
complexes …………………………………………………...…………………………...........241
Table 7.2: Aqueous phase calculated ligand binding energies in
kcal/mol, Mayer bond orders for
the U=O Bond and atomic Mulliken charges on uranium atoms in the
UO2Fn(H2O)5-n] 2-n
complexes obtained at the ADF/ZORA/TZP/BP86/COSMO level
………………………….247
Table 7.3: Calculated relative energies (kcal/mol), frontier gaps
(eV) and structural features of
the two orientations of the [UO2F5] 3-
complex in the hydrophobic cavities of tetrabrachion...256
Table 7.4: Computed structural parameters cisplatin in the gaseous
and aqueous phases and of
two randomly selected orientations of cisplatin embedded in the
cavity two (largest cavity) of the
RHCC protein obtained using RECPs ………………………………………………………..260
xvi
Table 8.1: Calculated and experimental structural parameters of 1a
in the ferromagnetic triplet
electronic state and antiferromagnetic broken-symmetry state (in
parentheses) ……………...279
Table 8.2: Calculated reaction free energies (ΔG298, kcal/mol)
required to transaminate the
unoccupied amine site of several Pacman complexes with a cis-uranyl
group ………..……...284
Table 8.3: Calculated atomic charges on the uranium and oxo- atoms
of the uranyl Pacman
complex, UO2(Py)(H2L), and its pentavalent and reductively
oxo-functionalized derivatives
obtained at the PBE/AE/4-component level (and at the B3LYP/RECP
level) ……………….284
Table 8.4: The calculated and experimental structural parameters
(bond lengths in Å and angles
in degrees) for the monomer complexes
………………............................................................286
Table 8.5: The calculated and experimental structural parameters
(bond lengths in Å and angles
in degrees) for the dimer complexes
………………...………………………………………...288
Table 9.1: Calculated relative energies (kcal/mol) of the three low
energy structures of
(UO2)2(OH)5 - ………………………………………………………………………………….302
Table 9.2. Calculated bond lengths (Å) and bond orders of the low
energy structures of
(UO2)2(OH)5 - obtained at the B3LYP/TZVP level
…………………………………………...303
Table 9.3. Calculated IR vibrational frequencies (cm -1
) of the low energy structures of
(UO2)2(OH)5 - obtained at the B3LYP/TZVP level
…………………………………………...307
Table 9.4. Calculated bond lengths (Å), bond orders and vibrational
frequencies (cm -1
) of a few
gas-phase mono-nuclear uranium complexes obtained at the B3LYP/TZVP
level …………..310
Table 9.5. Relative energies (kcal/mol) of the low energy
structures of (UO2)2(OH)n 4-n
, (n=2, 3, 4
and 6) obtained at the DFT and ab initio levels
……………………………………………….315
xvii
CASPT2 CASSCF with 2 nd
-Order Perturbative Treatment of Dynamical Correlation
CCI Cation-Cation Interactions
CCSD(T) Coupled Cluster Singles and Doubles with Perturbative
Triples
CCSDTQ Coupled Cluster Singles, Doubles, Triples and
Quadruples
CI Configuration Interaction
IR Infra-Red
KS Kohn-Sham
xviii
MO Molecular Orbital
NBO Natural Bond Orbitals
NMR Nuclear Magnetic Resonance
PBE Perdew Burke and Enzerhof GGA Functional
PCM Polarizable Continuum Model
PUREX Plutonium URanium EXtraction
SCF Self Consistent Field
STOs Slater Type Orbitals
UEG Uniform Electron Gas
xix
Abstract
Of the many available computational approaches, density functional
theory is the most
widely used in studying actinide complexes. This is generally
because it incorporates electron
correlation effects and is computationally inexpensive for modestly
sized compounds.
The first chapter of this thesis is an introductory chapter in
which some basic concepts of
electronic structure theory are discussed. The rest of this thesis
is a compilation of several studies
of the structural and electronic properties of a range of actinide
compounds using predominantly
density functional theory. The performances of the basis
set/relativistic components as well as
the density functional component of theoretical calculations were
examined in Chapters 2 and 3
respectively. In Chapters 4, 5, 6 and 7, the electronic structures
and properties of actinide species
in the environment were explored. The speciation of actinyl
aquo-hydroxo species at increasing
pH values were studied in Chapter 4. In Chapter 5, the structural
and electronic properties of
uranyl peroxo complexes with other environmentally important
ligands were studied. The
adsorption of uranyl complexes to geochemical surfaces was studied
in Chapter 6. In addition,
the mechanistic pathways to the reduction of these complexes on
surfaces and alcohols were
examined. In chapter 7, the complexes formed by the uranyl moiety
with the aquo and fluoride
ligands were studied in gas and aqueous phases. The interactions of
uranyl pentafluoride with a
protein were examined using a hybrid QM/MM approach. Overall these
studies (Chapters 4, 5, 6
and 7) provided valuable insights into the speciation and reduction
of actinide species in the
environment. In Chapter 8, the properties of novel pentavalent
uranium complexes were studied
using density functional theory. These complexes have promising
roles in the retardation of
uranium, via U(VI)-U(IV) reduction, in the environments of nuclear
storage repositories. In
xx
Chapter 9, the existence of cation-cation interactions in an
hexavalent bis-uranyl hydroxo
complex was examined using density functional theory and
wavefunction methods. In Chapter
10, a summary of the works compiled in this thesis is presented.
Future directions for work on
the chemistry of actinide complexes were also included in this
chapter.
xxi
Acknowledgement
I would like to thank my supervisor, Dr Georg Schreckenbach, for
the opportunity to
work under him towards this degree over the last few years. His
patience, support,
encouragement, friendly approach and critical thinking have been
invaluable along the way to
this degree. In the same vein, I would like to thank the members of
my advisory committee, Dr
Mario Bieringer, Dr Peter Budzelaar and Dr Mostafa Fayek for their
help, advice, criticism and
support during the work towards this degree.
I am indebted to all the present and former members of the
Schreckenbach group at the
University of Manitoba. The postdoctoral and research fellows,
Grigory Shamov, Qing-Jiang Pan
and Abu Asaduzamman were very helpful during the early stages of my
doctoral program. Their
help with the various projects I worked on as well as on-going
collaboration after their departure
from the group is highly appreciated.
I would like to thank all members of my family. Without the
encouragement of my loving
wife, Sarah Odoh, my sister, Mary Odoh, and my parents (Godwin and
Caroline Odoh), it would
not have been possible to complete this degree. I am eternally
grateful to my brothers, Emmanuel
and Daniel Odoh for their support and understanding.
1
1.1 The actinides and their uses.
The actinides or actinoids are the elements with atomic numbers
from 89 to 103 on the
periodic table of elements. 1-2
They are named after the first member of this series, actinium.
The
position of these elements on the periodic table is shown in Figure
1.1. The actinide series
correspond to filling of the 5f shell (mostly) and the actinides
are therefore f-block elements. The
only exception to this is lawrencium which is a d-block element. 3
The other members of the f-
block are the lanthanides. These are the elements from lanthanum to
lutetium which possess
gradually filled 4f shells.
Of the actinides, only uranium and thorium are found in substantial
quantities in nature.
Protactinium and actinium are also found in nature. All the other
actinides are artificial elements
produced through various nuclear reactions of primordial uranium.
In terms of abundance,
uranium and thorium exist at average concentrations of about 2-4
and 6 parts per million (ppm)
in the earth crust. Examples of thorium minerals are thorianite
(ThO2, 88% Th), thorite (ThSiO4,
72% Th) and brabantite (CaTh(PO4)2, 50% Th). The most common
uranium ore is uraninite
(UO2, 88% U). Others include Rutherfordine (UO2(CO3), 72% U) and
schoepite
[(UO2)8O2(OH)12.12H2O, 73% U]. Np, Pu, Cm, Bk and Cf are sometimes
formed by natural
transmutation of uranium ores and so are found in minute quantities
in these minerals. 4
The actinides have found their greatest applications in the
production of energy via
controlled nuclear fission and in the production of nuclear
weapons. These uses are underpinned
2
Figure 1.1: The actinides in the periodic table of elements.
5
by the radioactive behavior of the actinide elements. Radioactivity
is the ability of an unstable
nucleus to lose energy and decay to other nucleus/nuclei by
emitting ionizing radiation. Although
all isotopes of the actinides are radioactive, the 235 isotope of
uranium is the most commonly
used fissile material in nuclear reactors. Bombardment of this
isotope with a neutron leads to its
fragmentation into smaller nuclei and emission of 2-3 other
neutrons. As a result of the release of
more neutrons than were used in the initiation process, the fission
of 235
U becomes self-
sustaining after a critical mass (about 52 kg) is attained. This
chain-reaction of neutron-induced
fission of 235
U is controlled in nuclear reactors. The heat produced is used to
generate electricity.
Currently, about 7% of total global energy consumption and 14% of
global electricity
consumption is produced from nuclear reactors using some
fissionable actinide isotope. 6
3
There are however significant drawbacks to the use of the actinide
elements in nuclear
reactors to generate energy. 7 Firstly, significant portions of the
nuclear fuel used in fission
reactors are not consumed and end up as waste, generally a mixture
of U, Pu and other actinides.
Some constituents of the radioactive waste, especially fissile
plutonium, can be separated and
reused in nuclear weapons and other reactors. The plutonium uranium
extraction process,
PUREX, is one such method for extracting fissile actinides from
spent nuclear fuel. It is however
the case that spent fuel and nuclear waste are highly hazardous and
toxic to living things as well
as the environment. There is therefore a need to keep nuclear waste
and spent fuel from
contaminating the environment. The half-life of a radionuclide
(radioactive nucleus) is the period
of time it takes it to lose half of its radioactivity. The
actinides with very long half-lifes found in
nuclear waste pose strong challenges to waste storage, disposal and
management strategies. 232
U
has the shortest half-life (68.9 years) of the uranium isotopes
while 238
U has the longest half-life
(4.5 billion years). 237
Np and 239
Pu are also found in nuclear waste and have half-lifes of about
2
million and 24,000 years respectively. The rather long periods of
time needed for these
radioisotopes to lose their radioactivity, relative to the average
human life span, means that it is
well nigh impossible to prevent eventual dispersal into the
environment, waste management
approaches such as geologic disposal and transmutation
notwithstanding. For this reason, it is
very important that we have a clear understanding of the dispersal
and migration of actinide
elements, their speciation in the environment, their interaction
with abiotic surfaces as well as
with biotic organisms and the mechanisms behind their toxicity to
animal and plant life. 8 There is
a need to understand the overall chemistry occurring at already
contaminated sites (that are here
and now).
4
The actinides are also used in coloring glasses and ceramics (UO2)
9 , smoke detectors (Am
as alpha emission source) 10
, and gas mantles (Th). 239
Pu is extensively used in the nuclear
weapons industry while 238
Pu is used in heart pacemakers and deep-sea diving suits as a
source
of energy. 11
It was used as heating source for the astronauts who participated
in the Apollo space
missions. It is particularly suited for these roles as it emits
relatively harmless alpha particles.
239 Pu is now being used as a nuclear fuel in fast-breeder
reactors. Depleted uranium is used in
making battle armors and projectiles. The ability of actinide
elements and their compounds to
efficiently catalyze reactions is under continuing
investigation.
1.2 Chemical properties of the actinides
There have been a large number of studies examining the chemical
properties of the
actinide elements and their compounds. For the sake of brevity we
here focus on their oxidation
states and electronic configurations. The electronic ground state
configurations of the actinide
elements are given in Table 1.1. The 5f, 6d and 7s electrons are
close in energy to each other as a
result of the relative destabilization of the 5f orbitals due to
relativistic effects. The implication
of this is that the actinides can have a variety of oxidation
states as any number of ionized
electrons can be removed from the energetically close valence
energy levels. The various
oxidation states existing for each of the actinide elements are
shown in Table 1.2. The presence
of multiple oxidation states for the actinides has significant
ramifications to their speciation in
the environment as the stability and hydrolytic behavior of each
actinide elements differ for
different oxidation states. This is particularly common for the
light actinides (U, Np and Pu)
which in some cases exhibit more than one oxidation state in the
same solution. As expected, the
stabilities of the different oxidation states depend on the
electronic configuration (and resulting
stability) of the resulting ion. For example, the +2 oxidation
state is known to be only transiently
5
Table 1.1: Electronic ground state configurations of the actinide
elements. The noble gas core
structure of radon, [Rn], is used to depict the electronic
configurations. The atomic number, Z, of
each element is also given. 1
Element Z Configuration Element Z Configuration
Actinium 89 [Rn]6d 1 7s
2 Berkelium 97 [Rn]5f
9 7s
2 Californium 98 [Rn]5f
10 7s
1 7s
1! 7s
1 7s
12 7s
1 7s
13 7s
2 Nobelium 102 [Rn]5f
14 7s
2 Lawrencium 103 [Rn]5f
14 6d
1 7s
1 7s
2
stable for all the actinide elements (with the exception of
Nobelium, due to the presence of a
stable fully filled 5f sub-shell). Similarly the half-filled 5f 7
electronic configuration implies that
the +4 oxidation state is most stable for Berkelium. The different
oxidation states for these
elements result in a rich redox chemistry and consequently colorful
chemistry in solution. For
example aqueous solutions of uranium in the +3, +4 and +6 oxidation
states have visible colors
of brown-red, green and yellow respectively. The +4 and +6
oxidation states are most important
for uranium. They form the oxides, UO2 and UO3 respectively. The +5
oxidation state of
uranium is generally unstable as it disproportionates to the +4 and
+6 oxidation states. 12 The
different actinide oxidation states also generally have different
solubility and stability in aqueous
6
Table 1.2: The various oxidation states of the actinide
elements.
Actinium 3 Berkelium 3, 4
Thorium 3, 4 Californium 2, 3, 4
Protactinium 3, 4, 5 Einsteinium 2, 3
Uranium 3, 4 ,5 ,6 Fermium 2, 3
Neptunium 3, 4 ,5 ,6 ,7 Mendelevium 2, 3
Plutonium 3, 4 ,5 ,6 ,7 Nobelium 2, 3
Americium 3, 4 ,5 ,6 Lawrencium 3
Curium 3, 4, 5
environments. 8 Uranium in the +6 state, U(VI), is generally more
soluble in water than the +4
oxidation state, U(IV), which tends to form insoluble precipitates.
To deter contamination of the
aquatic environment and the ecological zones of cities and
countries by radioactive uranium
from spent fuel or mill tailings, it is preferable that U(VI)
compounds are reduced to the +4
oxidation state. This is because the U(IV) compounds are
significantly less soluble in water and
therefore migrate more slowly than U(VI) species and are more
amenable to retardation and
deposition strategies. 8 An interesting example of the role the
differing oxidation states play in the
environmental chemistry of the actinide elements has been described
by Runde. 8 Plutonium is a
rather problematic element in the waste isolation pilot plant,
WIPP, depository, in New Mexico,
USA. This is because the high chloride soil content of the
depository preferentially stabilizes
7
Pu(VI) which is more soluble (and thus more environmentally mobile)
than Pu(IV). In contrast,
the alkaline waters of the Yucca Mountain nuclear waste storage
site dissolve Np about a
thousand times more easily than Pu. As such Np is the problematic
element at Yucca Mountain,
Nevada, USA. 8
The +3 and +4 actinide ions exist as discrete ions which are
hydrated in solution as
An(H2O)n +3/+4
species. The +5 and +6 oxidation states however hydrolyze water in
aqueous
solutions to form actinyl, AnO2 + and AnO2
2+ ions respectively. The An
+3/+4 and AnO2
+/2+ ions
form a large number of coordination complexes by binding various
types of ligands. This is due
to the hard acid nature of these ions. They form strong complexes
with hard bases such as
anionic or oxygen donating ligands. Examples of such complexes are
thorium nitrate
(Th(NO3)4.5H2O), actinide halides (AnX4, An = Th, U, Np and Pu, X =
F, Cl and Br) and
uranium carbonate, UO2(CO3)3 4-
and AnO2 +/2+
ions depend on the size and shape
(steric effect) of the ligands. For example, the effective
coordination numbers for aquo ligands to
these ions are 8-10 and 9-12 for the trivalent and tetravalent ions
respectively and 4-5 and 5-6,
especially for the aquo ligand, for the pentavalent and hexavalent
actinyl species respectively.
The aquo ions for U(VI) and U(IV) with coordination numbers of 5
and 8 respectively are shown
in Figure 1.2.
As uranium is the most abundant and most widely used actinide
element, a larger
(compared to the other actinides) proportion of experimental
studies have focused in
synthesizing and characterizing its complexes. Although there has
been recent progress in the
synthesis and characterization of diverse uranium complexes, there
are still a lot of experimental
8
challenges to our understanding of the behavior of these elements
and their compounds in the
environment. A few of them are: 1) the need for difficult handling
techniques as a result of the
Figure 1.2: The aquo complexes of U(VI) and U(IV).
toxicity and radioactivity of the actinide elements and the
relative scarcity of the actinide
elements. This is especially true for the trans-uranium elements.
In many cases, several different
oxidation states might be simultaneously present in solution
further complicating the work of
experimental chemists. In contrast, computational studies are
relatively cheap, safe and can be
used to either complement available experimental data or bridge any
gaps in our understanding
of the properties of actinide complexes.
9
1.3.1 The Schrödinger equation
All theoretical approaches employed in this thesis begin with a
formulation of the time-
independent adiabatic approximation of the Schrödinger equation.
13
In these methods, the
Schrödinger equation is essentially recast to take advantage of the
Born-Oppenheimer (BO)
approximation 14
which allows the motion of the nuclei and electrons to be
uncoupled. The
physical meaning of the BO approximation is an assumption that the
nuclei moves at negligible
speeds compared to the electrons. The nuclei are assumed to be
stationary in comparison to the
fast moving electrons. This separation of the nuclear and
electronic motion allows for an easier
evaluation of the electron-nuclei, nuclei-nuclei interactions as
well as the nuclear kinetic
energies.
(1.1)
(1.2)
(1.3)
(1.4)
Vnuclei is constant at each molecular geometry (BO approximation)
and so can be
removed. This allows us to recast Equation 1.1 into the electronic
Schrodinger equation,
Equation 1.3. The variational minimization of the energy results in
the ground state
wavefunction, Ψ and the lowest electronic energy, E, obtained
during the variational
minimization depends parametrically on the nuclear positions. The
components of Equation 1.4
are the kinetic energy of the electrons, the electron-nuclei
interaction potential and the electron-
10
electron interaction potential. The minimization of the energy,
complete solution of the
Schrödinger equation, is however not tractable except for the
smallest systems such as H2, H2 + ,
and He + . The reason for this is the 3N-dimensional nature of the
electronic wavefunction, where
N is the number of electrons. The electron-electron interaction
part, , of the
Hamiltonian is also very difficult to evaluate.
1.3.2 The variational principle and electronic basis sets.
Examination of Equation 1.3 shows us that it has an infinite number
of solutions.
Assuming we are working with the electronic Schrodinger equation,
the electronic energy
obtained with an arbitrary wavefunction can be written as in
Equation 1.5. The solution that
yields the lower bound to the energy, Eel, yields the correct
ground state wave-function. The
lower bound to the energy is labeled as E0. The variational
principle in lay-man terms simply
states “if there are two wavefunctions for a system, the one that
produces the lower energy better
represents the ground state wavefunction of the system and in fact
the true ground state
wavefunction yields the lowest energy, E0”. According to the
variational principle, if the exact
ground state wavefunction is used in Equation 1.5, the lowest
electronic energy will be obtained.
Thus Equation 1.5 can be generalized to Equation 1.6.
11
It is immediately obvious that to get a given value of Eel as
described in Equation 1.6, we
need a trial wavefunction, its quality (seen as Eel – E0)
notwithstanding. For atomic and
molecular systems, electronic trial wavefunctions need to fulfill
two important properties. Firstly
as electrons are fermions, the trial wavefunction must be
anti-symmetric under particle
interchange.
And secondly, no two identical electrons (or fermions in general)
can occupy the same quantum
state simultaneously. This is called the Pauli Exclusion Principle.
The Slater determinant is an
expression that conforms to these two conditions for
multielectronic systems. 15
It is written as a
determinant consisting of several orthonormal spin-orbitals each of
which describe the position
and spin of an electron.
There are generally two classes of functions used to represent
atomic spin-orbitals in
modern electronic structure theory. 16
The first are called Slater type orbitals (STOs) which are
modeled using the exponentially decaying electronic distribution (e
-ζr
) of the hydrogen atom. As
they correspond to a real atom, they accurately describe the cusp
condition at the atomic center.
There are however significant computational challenges to
calculating the integrals of an
exponential decay function. In most cases, numerical solutions to
the integrals of these functions
are needed. The cusp at the atomic center is also difficult to
handle computationally. The
Gaussian type orbitals (GTOs) are a more tractable class of
orbitals. Such orbitals possess
12
analytical solutions to their integrals. The disadvantages of using
GTOs are: 1) wrong cusp
conditions and 2) overall poor long range behavior as they decay
much faster than STOs, e -αr2
instead of e -ζr
. A widely used approach to circumvent these disadvantages is to
use several GTOs
to represent a single basis function according to Equation 1.9. The
coefficients, Ci, are fit to
ensure agreement with the radial electronic distribution of the
hydrogen atom.
In molecular systems, the atomic basis sets are combined to
represent a molecular orbital
(MO) as shown in Equation 1.10. This is called the Linear
Combination of Atomic Orbitals
principle (LCAO). The coefficients, Si, describe the contribution
of each atomic orbital to the
MO. Prior to going forward, we note that the constituent atomic
orbitals of an MO can contain
one basis function (either an STO or GTO). These types of basis
sets are said to be of single-ζ
quality. Those containing two, three and four basis functions per
atomic orbital are said to be of
double-ζ, triple-ζ and quadruple-ζ basis sets. Polarization
functions (basis functions with higher
angular momentum) as well as diffuse basis functions can be added
to each atomic orbital to
respectively allow for a better description of electron correlation
and anionic systems or excited
electronic states. In some cases, the basis function is partitioned
into a core and valence part.
This is generally rooted in the idea that core electrons are
generally chemically inactive and not
involved in bond formation or ionization processes. A more complete
explanation of the
variational principle and the other concepts briefly described here
can be found in most modern
computational chemistry textbooks. 16-17
1.3.3 The Hartree-Fock method and Post Hartree-Fock
approaches.
At the Hartree-Fock level, a Fock operator consisting of
one-electron kinetic energy, the
nucleus-electron interaction potential and the Hartree-Fock
potential is defined, Equation 1.11.
The Hartree-Fock method is the direct result of applying the
variational method to the Slater
determinant, Equation 1.8, in the electronic Schrödinger equation,
Equation 1.1. The first two
terms of Equation 1.11 (the electron kinetic energies and
nucleus-electron interaction potentials)
are one-electron operators describing the motion of an i th
electron in the field of the nuclei. These
are combined into the term of Equation 1.12. The Hartree-Fock
potential consists of the
Coulomb (j) and exchange ( j) operators. It is a two-electron
operator that describes inter-
electron repulsion. The former, j, describes the classical
Coulombic repulsion between the
electron and the j th
electron. The exchange operator, j, defines the electron exchange
energy
and it essentially switches the spin orbital of the i th
electron with that of the j th
electron. The
Hartree-Fock Hamiltonian is then described as the sum of the Fock
operators for all electrons in
the system, Equation 1.14. 18-19
This equation allows us to write the Hartree-Fock (HF) energy
as
14
It becomes immediately obvious that HF theory, from the definition
of the Hartree-Fock
potential, is a mean-field approximation in which each electron
moves in an average field
generated by the remaining (n-1) electrons. There are two very
important deficiencies to the HF
solution with a single Slater determinant. The first relates to the
difference between the
instantaneous electron-electron interaction of the real system and
the mean field interaction
experienced by each electron in HF theory. This is called dynamical
correlation. The second
deficiency is labeled as non-dynamical correlation and relates to
deficiencies caused by the use
of a single Slater determinant to describe the system.
Non-dynamical correlation is particularly
large for systems with near-degenerate energy levels. As such the
true wavefunction of such
systems consist of several coefficient-weighted Slater
determinants. The sum of the dynamical
and non-dynamical correlation is called the correlation energy and
is defined as the difference
between the ground state electronic energy and the HF energy.
It is important to note that although the contribution of electron
correlation to the total
electronic energy is small (~1%), its omission in most cases leads
to large errors in calculated
structural and electronic properties as well as reaction energies.
There are various approaches to
improving the HF approach. The only post Hartree-Fock approaches
employed in this work are
the second-order Møller-Plesset perturbation theory, MP2 and the
coupled cluster singles and
doubles and perturbatively included triples, CCSD(T),
approaches.
15
the unperturbed Hamiltonian, , is the sum of
the one-electron Fock operators in Equation 1.13, and the
Hartree-Fock wavefunction is an
eigenfunction of 0 which yields an eigenvalue equal to the sum of
the one electron energies of
the occupied spin orbitals. The electron correlation (dynamical
correlation as the single Slater
determinant is still employed) is described as a perturbation, ′ ,
to
0 .
Essentially a subset of the general time-independent perturbation
theory of Rayleigh and
Schrödinger 22
, Møller-Plesset perturbation theory assumes that all Slater
determinants
corresponding to the excitation of electrons from the occupied to
the virtual orbitals are also
eigenfunctions of 0 with an eigenvalue equal to the sum of the one
electron energies of their
occupied spin orbitals. The wavefunction and energy can be expanded
in terms of the
perturbation.
Substitution of these equations into the electronic Schrödinger
equation yields a series of
equations conforming to the general formula: and .
The sum of and describes the HF energy (combination of the
one-electron energy and the
Hartree-Fock potential). The various Møller-Plesset perturbation
(MP) approaches are named
according to the degree of correlation correction included with the
HF potential. For example, in
MP2, is included. Lastly, it should be noted that the higher orders
of the
MP n approaches are not necessarily convergent with respect to
total energies.
16
16
In coupled cluster theory, an infinite exponential cluster operator
acts on the Hartree-
Fock wavefunction to generate a full configuration interaction, CI,
solution accounting for
electron correlation, Equation 1.19. 23
The coupled cluster approaches are labeled according to
the level at which the expansion coefficients, , are truncated,
Equation 1.20. 24
As an example,
truncation at the single and double excitations level leads to the
coupled cluster singles and
doubles approach (CCSD). The most popular coupled cluster approach
involves the inclusion of
triple excitations into the CCSD wavefunction in a perturbative
(Møller-Plesset) manner,
Equation 1.16. This method is called the CCSD(T) approach. The
CCSD(T) method is often
referred to as the gold standard of computational chemistry as it
has been shown to yield highly
accurate (< 1 kcal/mol) reaction energies, transition state
barriers and structure. 24-25
It is also a
compromise between computational expense and the more accurate
CCSDTQ and higher order
coupled cluster approaches.
To briefly review, the major deficiency of HF theory is the lack of
an appropriate
treatment of the instantaneous interaction between electrons as
well as the single reference nature
of the Slater determinant employed. This neglect of electron
correlation and mean field
approximation to inter-electron repulsion has dramatic effects on
calculated bond energies,
vibrational frequencies and bond lengths in molecular species.
16
The MP n and coupled cluster
approaches become increasingly expensive at higher orders. The
energy convergence of the MP n
methods is in many cases oscillatory at increasingly higher orders.
These approaches also
experience significant spin contaminations when employed for open
shell (unrestricted HF
17
wavefunction) systems. The coupled cluster techniques are only
computationally feasible (at the
current moment) for the smallest of molecules. They also require
basis sets of at least triple-ζ
quality.
1.3.4 Density Functional Theory
Most of the calculations in this thesis were carried out under the
framework of density
functional theory (DFT). Although it has its roots in the much
earlier Thomas-Fermi model 26-27
and the work of Slater on the Xα exchange functional 15, 28
, modern DFT is based on the two
fundamental Hohenberg-Kohn (HK) theorems. 29
The first HK theorem states that the ground
state electron density, ρ(r), uniquely determines the external
potential V(r). The determination of
V(r) allows for the exact formulation of the Hamiltonian and thus
the wavefunction. The second
HK theorem states that the ground state energy can be obtained
variationally. This is due to the
fact that any new density generates a new external potential
leading to a new wavefunction. The
energy, a functional of the density, is the sum of the external
potential, , the kinetic
energy, and electron-electron interaction energy, , Equation 1.21.
The
Eee term contains both the classical Coulombic interactions and the
non-classical electron-
electron interactions.
Kohn-Sham DFT is the most widely used form of DFT. The basic
assumption to this
framework is the stipulated existence of a system of
non-interacting electrons with exactly the
18
same density as the system of interest, one with interacting
electrons. 30
For the system of non-
interacting electrons, the Eee term of Equation 1.21 by definition
becomes zero. An analogous
equation to the Fock equations of HF theory (Equations 1.11-1.14)
allows us to construct Kohn-
Sham spin orbitals. in Equation 1.23 is the energy of the spin
orbitals generated for the
non-interacting system. Equation 1.24, is an eigen-value equation
for Kohn-Sham particles and is
reminiscent of the Fock equations at the Hartree-Fock level,
Equation 1.12.
The one-electron Hamiltonian for the Kohn-Sham particles is given
as
The effective external potential is defined as
The energy of the Kohn-Sham system is then given as
The exchange-correlation term, , is a composite term containing: 1)
the kinetic energy
of a system of interacting electrons minus the kinetic energy of a
system with non-interacting
electrons with exactly the same density and 2) the
electron-electron interaction (total electron
exchange and correlation) minus the classical Coulomb interaction.
Working backwards, we can
19
see that the Schrödinger equation has essentially been recast with
energy as functional of density.
If the exact exchange-correlation term is known, a full solution to
the Schrödinger equation will
have been obtained. Although a large number of possible
exchange-correlation functionals have
been proposed and tested over the years, 31-36
the exact nature of this functional remains
unknown. This functional is often split into the exchange
functional and the correlation
functional.
The earliest known exchange-correlation functional is called the
local density
approximation (LDA) and is based on the fictitious system of a
uniform electron gas, UEG. This
system has an infinite number of electrons and uniform density all
throughout. The exchange
energy of a UEG is a functional of its density, Equation 1.29. The
correlation part of the
exchange-correlation functional was parameterized using highly
accurate Monte Carlo
calculations on the UEG. This work was done by Ceperley and Alder.
37
Modern modifications
such as that of Vosko, Wilk and Nusair are available in most modern
ab initio software
packages. 38
Testing of the performance of the LDA functional for actinide
molecules reveals that
it generally leads to bond lengths that are slightly shorter than
experimental values as well as
reaction energies that deviate significantly from the experimental
values. LDA is however
particularly suited for metallic periodic systems where the
electron density changes only very
slowly. 16-17
20
The most common modifications to the LDA functional involve
attempts to correct for
rapidly changing electron densities found in molecular systems. The
generalized gradient
approximation, GGA, functionals were the first corrections to find
widespread usage for
molecular systems. In these functionals the gradient of the density
is included in the formulation
of the exchange-correlation functional, Equation 1.30. The term is
the gradient
parameter and provides a better description of the electron
exchange in regions where there are
changes in the electron density. The GGA functionals provided vast
improvement on the
agreement between the calculated ionization, atomization and
binding energies and the
experimental values. 16
Rigorous tests have however found that they generally tend to
slightly
overestimate the lengths of the bonds in actinide complexes.
39-43
The PBE functional is a non-
empirical GGA functional and was widely used in the works compiled
in this thesis. 44-45
The next stages of modification to the exchange-correlation
functional encompass the
hybrid and meta-GGA functionals. Additional information regarding
the density is included as
the Laplacian of the density and the non-interacting kinetic energy
in the meta-GGA functionals.
No meta-GGA functionals were employed in the works compiled in this
thesis. Regarding the
hybrid functionals, the gradient corrected GGA functionals are
combined with explicit Hartree-
Fock exchange. An example of the general mixing of GGAs with
Hartree-Fock exchange as seen
in the Becke three parameter functionals is shown in Equation 1.31.
This combination is based
on the adiabatic connection formula which allows us to write the
exchange-correlation formula
as a combination of the Hartree-Fock exchange and some DFT
exchange-correlation functional.
21
The most widely used functional in the works compiled in this
thesis and all of computational
actinide chemistry in general is the B3LYP functional. 46-47
It combines some portion of the
Hartree exchange with the B88 exchange and LYP correlation
functionals using the formula:
The B88 exchange functional of Becke and the popular LYP
correlation functional of Lee Yang
and Parr are the GGA functionals employed in the B3LYP functional.
Extensive testing by
various authors have shown this particular functional to be suited
for calculating the structural
properties of actinide complexes, their vibrational frequencies as
well as their reaction
energies. 39, 41-43, 48
Other examples of hybrid functionals include B3PW91 and BHandH.
16
Other functionals employed in this work are the long-range
corrected hybrid functional,
CAM-B3LYP 49
, and the B3LYP functional
with dispersion corrections using Grimmes third scheme, B3LYP-D3.
33
1.3.5 Relativistic effects
The time-independent Schrödinger equation 13, 22
is only valid for non-relativistic systems.
As the atomic number, Z, increases in heavier nuclei, the speeds of
the core-electrons approach
the speed of light. At such speeds, relativistic effects become
important in the electronic structure
of the atoms, ions and compounds of heavy nuclei. Generally the
major effects in molecular
systems are propagated by the contraction and stabilization of s
and p orbitals as well as the
expansion and destabilization of d and f orbitals. 50-52
Relativistic effects increase according to
Z 2 /c
2 down the periodic table, where c is the speed of light, 137 in
atomic units. The Dirac
equation, Equation 1.32, was formulated in 1929 to describe the
one-electron energies of
relativistic systems 53
. In the Dirac equation,
22
α and β are 4×4 matrices, Equations 1.33-1.35, V is the external
potential and the wavefunction is
a four component column spinor, Equation 1.36, containing the large
component, ΨL, and the
small component, ΨS. The large and small components respectively
describe the positive and
negative energy solutions found on either side of 2mc 2 . The small
component solutions
correspond to positrons while the large component solutions
correspond to electrons. The up or
down arrows in Equation 1.36 indicate the particle spin of the
electrons and positrons. The
coupling between the positronic and electronic parts of the
wavefunction can be removed
through the Foldy-Wouthuysen (FW) transformation. 55
The Pauli and Zeroth order regular
approximation (ZORA) Hamiltonians were obtained by first order FW
transformation of the
Dirac-Coulomb-Breit Hamiltonian expanded in a series of c -2
and E/(2mc 2 -V).
56-57 Calculations
in which the spin components are projected out of the
four-component Dirac equation are known
as scalar-relativistic calculations while those in which the
small-component is projected out are
two-component approximations to the full Dirac equation. In these
sets of equations (Equations
1.32-1.36), I is the two-dimensional identity matrix, σ represents
the Pauli spin matrices in a
compact form and p is the momentum.
23
one of the ZORA 56-57
, the Douglas-Kroll-Hess 58-59
. The ZORA and ECP approaches were employed in the
calculations summarized in this thesis. We however here focus on
discussing the use of
relativistic ECPs as they were used in nearly all the studies
included in this work.
The realization that the core electrons unlike their valence
counterparts are chemically
inert and not involved in bond formation implies that they can be
completely frozen (frozen-core
approximation) or replaced by an effective core potential (ECP)
without any massive loss in
accuracy. The overall Hamiltonian used in the ECP calculations is
transformed into Equation
1.37 below. In this equation, contains the Coulomb and exchange
operators for the
valence electrons and potential to account for the core electrons.
The kinetic and electron-
electron correlation terms are completely non-relativistic.
The design of ECPs usually starts from all-electron basis set
calculations on an atom of
the desired element. The design could either be for
non-relativistic ECPs in which case an HF
(and in some cases post-HF) wave-function is employed or for
relativistic ECPs in which some
approximation to the Dirac equation (in most cases, the quasi
relativistic Pauli Hamiltonian) is
used. For the latter, all relativistic effects are embedded in the
Vpp term of Equation 1.37. The
designers of either flavor of ECPs however face certain challenges:
62-63
A) Would the ECP be
24
constructed such that it replicates the shapes and energies and
shapes of the orbitals (shape-
consistent ECPs) or would it be an energy-consistent ECP which
would replicate some electronic
property (ionization potentials, electron affinities or electronic
spectra) of the atom? The energy-
consistent ECPs are the most widely used variety as a result of the
availability of experimental
data regarding the electronic properties of most atoms. B) Where is
the core-valence boundary?
In other words, which orbitals should be placed in the core region
to be replaced by the ECP and
which should be in the valence region? For the actinide elements,
this particular question is non-
trivial given the semi-core nature of the 6s and 6p orbitals as
well as the participation of the 5f
orbitals in bond formation.
There are two commonly used types of energy-consistent ECPs used in
computational
studies of actinide systems. 64-66
The design of the small-core (SC) ECPs is such that 60
(principal
quantum number, n < 5 shells) core electrons are represented
with a pseudopotential while the
remaining (n ≥ 5 shells) electrons (30 for thorium, 32 for uranium
as examples) are represented
by valence basis sets. For the second variety, the large-core (LC)
ECPs, 78 core electrons are
replaced by a pseudopotential. As the core-region covered by the
pseudopotential is larger for the
LC-ECPs, they provide even greater time savings than the SC-ECPs.
However, it has been
shown that SC-ECPs generally tend to provide better agreements
between experimental and
calculated structural parameters and reaction energies. In Chapter
2 and works by other authors,
the performance of the SC-ECPs and LC-ECPs designed by
Stuttgart-Cologne group relative to
experimental results (of the structure and reaction energies of
several uranium complexes) are
compared to those obtained using a full scalar relativistic
four-component approximation to the
Dirac equation. 41-43
1.3.6 Solvation Effects
The effect of a solvent environment on the nature,
speciation/chemical form and spectra
of actinide complexes can be significant. For example for the
meta-stable U(V) complexes, an
aqueous solvent is sufficiently oxidizing to prevent their
isolation and characterization. 12, 67-71
Experimental NMR studies of UO2 2+
in alkaline solution have shown the existence of fast
oxygen exchange processes. 48, 72-74
It should be noted that the behaviors of actinide species in
solutions are in no way uniform. For example, a comparison of the
speciation diagrams of UO2 2+
and PuO2 2+
, indicated that although the pentaaquo complex is dominant for
both species in
highly acidic solutions, the uranyl moiety forms trinuclear (in
addition to binuclear) species,
[(UO2)3(OH)5] + and [(UO2)3(OH)7]
2- at modest to high pH values while its plutonyl counterpart
forms mainly the binuclear species, [(PuO2)2(OH)2] 2+
with negligible concentrations of trinuclear
plutonyl complexes, even at high pH values. 75-77
The inclusion of solvent effects, when needed,
is therefore very important in modern computational actinide
chemistry.
One approach for describing the effect of a solvent environment on
the structure,
speciation and electronic properties of actinide complexes involves
adding a large number of
solvent molecules around the solute molecule. As an example, to
study the uranyl ion in aqueous
solution, the UO2 2+
ion could be surrounded by a cubic box containing water molecules
with an
overall density of about 1.0 g/cm 3 . Vibrational frequency
analyses and other calculations are then
carried out on this solute-solvent box after the structural
optimizations. This explicit approach to
modeling solvation effects can be extremely computationally
demanding as the number of
solvent molecules increase. In addition, the introduction of so
many solvent molecules implies
the existence of a large number of possible minima and transition
state structures. This further
complicates any search for the global minimum structure. 16
26
For the implicit solvation models, 78-82
a polarizable continuum or conductor-like model
with electrostatic and entropic properties that match that of the
desired solvent is employed. For
these solvation models, the size/shape of the solvent-excluded
cavity around the solute molecule
has to be defined. The molecular free energy in solution is
calculated as a sum of the
electrostatic, dispersion-repulsion and cavitation energy
contributions, Equation 1.38.
The cavity can be described such that each atom in a molecule is
represented by
individual spheres, individual cavity models, or for the united
atom models, in which the
hydrogen atoms are included in the spheres of the atoms to which
they are directly bonded. Most
software suites allow the user to specify the formation of the
cavity from various tesserae as well
as the radii of each cavity. The implementation of the polarizable
continuum solvation model 83-84
(PCM) in the Gaussian 03 suites of programs 85
was used in most of the work compiled in this
thesis. Gutowski et al. 86-87
have previously explored the suitability of this model and
other
solvation models for describing the effect of solvent environments
on the structures and
stabilities of actinide complexes. It has been shown that implicit
model calculations on AnO2 2+
and AnO2 + ions with a first solvation sphere containing 4-6
explicit water molecules accurately
reproduces the structure and properties of these ions in highly
acidic aqueous solutions. 41-42, 88
1.4 Organization of this Thesis
This thesis is written in a sandwich style agglomeration of several
manuscripts published
or submitted in peer-reviewed scientific journals during the course
of the doctoral program. The
over-arching aim of the dissertation is to further our
understanding of actinide chemistry by
using computational methods. In each chapter, we try to answer
particular questions regarding
27
the structure and properties of actinide complexes. The structural
and electronic properties as
well as the chemistry of actinide species in the gaseous, aqueous
and solid phases are examined
using theoretical calculations. In addition, the performances of
the various theoretical methods
used were also examined. Each chapter is followed by a list of
references. The basic outline of
this thesis is illustrated in Figure 1.3.
A brief introduction to actinide chemistry and the computational
methods used in this
thesis are presented in Chapter 1. In Chapter 2, we attempted to
benchmark the performance of
LC-ECP and SC-ECPs against an all-electron basis set four-component
scalar-relativistic
approach in the framework of DFT. This chapter essentially answers
the question: how much
accuracy can be obtained with ECP calculations? And how accurate
are DFT calculations
employing relativistic ECPs compared to those employing
all-electron electron basis sets with a
four-component approximation to the Dirac Hamiltonian?
In Chapters 3, 4 and 5, several questions regarding the aqueous
chemistry of actinide
complexes were studied using DFT calculations. The ability of DFT,
a single-reference theory to
accurately describe the structure of plutonium complexes is
confirmed in Chapter 3. It is very
important t